Showing papers in "Advances in Computational Mathematics in 1996"
TL;DR: A new discussion of the complex branches of W, an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containing W are presented.
Abstract: The LambertW function is defined to be the multivalued inverse of the functionw →we
w
. It has many applications in pure and applied mathematics, some of which are briefly described here. We present a new discussion of the complex branches ofW, an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containingW.
5,591 citations
TL;DR: Hartree has suggested two methods (methods I and II below) of evaluating approximate solutions of partial differential equations in two
Abstract: A, k, q are known constants. Equation (1) is of the type which arises in problems of heat flow when there is an internal generation of heat within the medium; if the heat is due to a chemical reaction proceeding at each point at a rate depending upon the local temperature, the rate of heat generation is often defined by an equation such as (2). The presence of the non-linear term q(Ow/at) in (1) and the empirical nature, in many cases, of the surface heat transfer function Hi (0), render the use of formal methods unsuitable. Now Hartree [3-5] has suggested two methods (methods I and II below) of evaluating approximate solutions of partial differential equations in two
845 citations
TL;DR: Three greedy algorithms are discussed: the Pure GreedyAlgorithm, an Orthogonal Greedy Algorithm, and a Relaxed Gre greedy Algorithm.
Abstract: Estimates are given for the rate of approximation of a function by means of greedy algorithms. The estimates apply to approximation from an arbitrary dictionary of functions. Three greedy algorithms are discussed: the Pure Greedy Algorithm, an Orthogonal Greedy Algorithm, and a Relaxed Greedy Algorithm.
559 citations
TL;DR: It is shown that the error in the finite element approximation is of optimal order for linear elements on a quasiuniform triangulation.
Abstract: This paper is concerned with the analysis of a finite element method for nonhomogeneous second order elliptic interface problems on smooth domains. The method consists in approximating the domains by polygonal domains, transferring the boundary data in a natural way, and then applying a finite element method to the perturbed problem on the approximate polygonal domains. It is shown that the error in the finite element approximation is of optimal order for linear elements on a quasiuniform triangulation. As such the method is robust in the regularity of the data in the original problem.
319 citations
TL;DR: An extension of the standard barycentric coordinate functions for simplices to arbitrary convex polytopes is described, the construction of a unique polynomial associated with that polytope, which generalizes a previous two-dimensional construction described by Wachspress.
Abstract: An extension of the standard barycentric coordinate functions for simplices to arbitrary convex polytopes is described. The key to this extension is the construction, for a given convex polytope, of a unique polynomial associated with that polytope. This polynomial, theadjoint of the polytope, generalizes a previous two-dimensional construction described by Wachspress. The barycentric coordinate functions for the polytope are rational combinations of adjoints of various dual cones associated with the polytope.
187 citations
TL;DR: In this article, a continuous version of spherical multiresolution is described, starting from continuous wavelet transform on the sphere, and it is shown that singular integral operators forming a semigroup of contraction operators of class (C0) (like Abel-Poisson or Gaus-Weierstras operators) lead in a canonical way to (pyramidal) algorithms.
Abstract: A continuous version of spherical multiresolution is described, starting from continuous wavelet transform on the sphere. Scale discretization enables us to construct spherical counterparts to wavelet packets and scale discrete wavelets. The essential tool is the theory of singular integrals on the sphere. It is shown that singular integral operators forming a semigroup of contraction operators of class (C0) (like Abel-Poisson or Gaus-Weierstras operators) lead in a canonical way to (pyramidal) algorithms.
137 citations
TL;DR: The complex representation of polynomial Pythagorean-hodograph curves allows the problem of constructing aC2 PH quintic “spline” that interpolates a given sequence of points p0,p1,...,pN and end-derivatives d0 anddN to be reduced to solving a “tridiagonal” system of N quadratic equations in N complex unknowns.
Abstract: The complex representation of polynomial Pythagorean-hodograph (PH) curves allows the problem of constructing a C 2 PH quintic "'spline" that interpolates a given sequence of points P0, Pt,.-., Pu and end-derivatives d o and du to be reduced to solving a "tridiagonal" system of N quadratic equations in N complex unknowns. The system can also be easily modified to incorporate PH-spline end conditions that bypass the need to specify end-derivatives. Homotopy methods have been employed to compute all solutions of this system, and hence to construct a total of 2 ~+~ distinct interpolants for each of several different data sets. We observe empirically that all but one of these interpolants exhibits undesirable "looping" behavior (which may be quantified in terms of the elastic bending energy, i.e., the integral of the square of the curvature with respect to arc length). The remaining "good" interpolant, however, is invariably a fairer curve-having a smaller energy and a more even curvature distribution over its extent-than the corresponding "ordinary" C 2 cubic spline. Moreover, the PH spline has the advantage that its offsets are rational curves and its arc length is a polynomial function of the curve parameter.
85 citations
TL;DR: It is proved that the minimum number of neurons in a neural network with a single hidden layer required in order to provide a mean approximation order of a preassignedε>0 to each function inW cannot be $$\mathcal{O}( \in ^{ - s} log(1/ \in ))$$ if a spline-like localization is required.
Abstract: Lets≥1 be an integer andW be the class of all functions having integrable partial derivatives on [0, 1]
s
. We are interested in the minimum number of neurons in a neural network with a single hidden layer required in order to provide a mean approximation order of a preassignede>0 to each function inW. We prove that this number cannot be
$$\mathcal{O}( \in ^{ - s} log(1/ \in ))$$
if a spline-like localization is required. This cannot be improved even if one allows different neurons to evaluate different activation functions, even depending upon the target function. Nevertheless, for anyδ>0, a network with
$$\mathcal{O}( \in ^{ - s - \delta } )$$
neurons can be constructed to provide this order of approximation, with localization. Analogous results are also valid for otherL
p
norms.
81 citations
TL;DR: A family of second-order,L0-stable methods is developed and analysed for the numerical solution of the simple heat equation with time-dependent boundary conditions and no oscillations are observed in the computed solutions.
Abstract: A family of second-order,L
0-stable methods is developed and analysed for the numerical solution of the simple heat equation with time-dependent boundary conditions. Methods of the family need only real arithmetic in their implementation. In a series of numerical experiments no oscillations, which are a feature of some results obtained usingA
0-stable methods, are observed in the computed solutions. Splitting techniques for first- and second-order hyperbolic problems are also considered.
68 citations
TL;DR: This paper suggests linear unlearning of examples as an approach to approximative cross-validation and shows that the generalization performance of the equally weighted ensemble predictor is identical to that of the network trained on the whole training set.
Abstract: The leave-one-out cross-validation scheme for generalization assessment of neural network models is computationally expensive due to replicated training sessions. In this paper we suggest linear unlearning of examples as an approach to approximative cross-validation. Further, we discuss the possibility of exploiting the ensemble of networks offered by leave-one-out for performing ensemble predictions. We show that the generalization performance of the equally weighted ensemble predictor is identical to that of the network trained on the whole training set.
54 citations
TL;DR: A Crank-Nicolson finite element Galerkin method and an algebraically-linear extrapolated Cranksville method for the numerical solution of a semilinear parabolic problem with nonlocal boundary conditions are formulated and analyzed.
Abstract: We formulate and analyze a Crank-Nicolson finite element Galerkin method and an algebraically-linear extrapolated Crank-Nicolson method for the numerical solution of a semilinear parabolic problem with nonlocal boundary conditions. For each method, optimal error estimates are derived in the maximum norm.
TL;DR: This paper deals with a class of non-stationary multiresolution analysis and wavelets generated by certain radial basis functions that satisfy the Littlewood-Paley identity and gives a detailed analysis of the time-frequency localization of these wavelets.
Abstract: In this paper, we deal with a class of non-stationary multiresolution analysis and wavelets generated by certain radial basis functions. These radial basis functions are noted for their effectiveness in terms of “projection”, such as interpolation and least-squares approximation, particularly when the data structure is scattered or the dimension of ℝ
s
is large. Thus projecting a functionf onto a suitable multiresolution space is relatively easy here. The associated multiresolution spaces approximate sufficiently smooth functions exponentially fast. The non-stationary wavelets satisfy the Littlewood-Paley identity so that perfect reconstruction of wavelet decompositions is achieved. For the univariate case, we give a detailed analysis of the time-frequency localization of these wavelets. Two numerical examples for the detection of singularities with analytic wavelets are provided.
TL;DR: In this paper, quadrature formulas on the unit circle are considered and algebraic properties are given and results concerning error and convergence established.
Abstract: In this paper, quadrature formulas on the unit circle are considered. Algebraic properties are given and results concerning error and convergence established.
TL;DR: In this article, a simple but fast algorithm is proposed that alternates between minimizing the number of misclassified points by translation of the separating plane, and a rotation of the plane so that it minimizes a weighted average sum of the distances of the misclassified objects to the separation plane.
Abstract: Given two finite point setsA andB in then-dimensional real spaceR
n
, we consider the NP-complete problem of minimizing the number of misclassified points by a plane attempting to divideR
n
into two halfspaces such that each open halfspace contains points mostly ofA orB. This problem is equivalent to determining a plane {x | x
T
w=γ} that maximizes the number of pointsx ∈A satisfying inx
T
w>γ, plus the number of pointsx ∈B satisfyingx
T
w<γ. A simple but fast algorithm is proposed that alternates between (i) minimizing the number of misclassified points by translation of the separating plane, and (ii) a rotation of the plane so that it minimizes a weighted average sum of the distances of the misclassified points to the separating plane. Existence of a global solution to an underlying hybrid minimization problem is established. Computational comparison with a parametric approach to solve the NP-complete problem indicates that our approach is considerably faster and appears to generalize better as determined by tenfold cross-validation.
TL;DR: The condition which ensures linear independence of the plane waves is investigated and can be used for proving structural uniqueness of neural networks and for implementing finite mapping by neural networks.
Abstract: Any number of linearly independent plane waves can be obtained by scaling, shifting and/or rotating a plane wave created from an activation function. We investigate the condition which ensures linear independence of the plane waves. In the case all the three procedures are combined, the linear independence can be proved by ad hoc methods for most of commonly used activation functions. The result can be used for proving structural uniqueness of neural networks and for implementing finite mapping by neural networks.
TL;DR: An analysis is provided which guarantees that the W-cycle and variable V-cycle multigrid algorithms converge with a rate of iterative convergence which can be bounded independently of the number of multilevel spaces.
Abstract: In this paper, we examine multigrid algorithms for cell centered finite difference approximations of second order elliptic boundary value problems. The cell centered application gives rise to one of the simplest non-variational multigrid algorithms. We shall provide an analysis which guarantees that the W-cycle and variable V-cycle multigrid algorithms converge with a rate of iterative convergence which can be bounded independently of the number of multilevel spaces. In contrast, the natural variational multigrid algorithm converges much more slowly.
TL;DR: Given a smooth functionu, the authors construct a piecewise polynomial functionυ ∈Cr (ℝ2) of degreen=2r forr even forr odd in each triangle ofτ6, interpolating derivatives ofu up to orderr at the vertices ofτ.
Abstract: Letτ be the triangulation generated by a uniform three direction mesh of the plane. Letτ
6 be the Powell-Sabin subtriangulation obtained by subdividing each triangleT ∈τ by connecting each vertex to the midpoint of the opposite side. Given a smooth functionu, we construct a piecewise polynomial functionυ ∈C
r
(ℝ2) of degreen=2r (resp. 2r+1) forr odd (resp. even) in each triangle ofτ
6, interpolating derivatives ofu up to orderr at the vertices ofτ.
TL;DR: A general result establishes the countability, and under stronger assumptions finiteness, of the set of functions giving rise to critical points of the quadratic loss function.
Abstract: This paper deals with nonlinear least-squares problems involving the fitting to data of parameterized analytic functions. For generic regression data, a general result establishes the countability, and under stronger assumptions finiteness, of the set of functions giving rise to critical points of the quadratic loss function. In the special case of what are usually called “single-hidden layer neural networks”, which are built upon the standard sigmoidal activation tanh(x) (or equivalently (1 +e
−x
)−1), a rough upper bound for this cardinality is provided as well.
TL;DR: This paper considers the development of a fourth order rational approximant to the matrix exponential function possessing real and distinct poles which readily admits a partial fraction expansion, thereby allowing the distribution of the work in solving the corresponding linear algebraic systems in essentially Backward Euler-like solves on concurrent processors.
Abstract: Time dependent problems in Partial Differential Equations (PDEs) are often solved by the Method Of Lines (MOL). For linear parabolic PDEs, the exact solution of the resulting system of first order Ordinary Differential Equations (ODEs) satisfies a recurrence relation involving the matrix exponential function. In this paper, we consider the development of a fourth order rational approximant to the matrix exponential function possessing real and distinct poles which, consequently, readily admits a partial fraction expansion, thereby allowing the distribution of the work in solving the corresponding linear algebraic systems in essentially Backward Euler-like solves on concurrent processors. The resulting parallel algorithm possesses appropriate stability properties, and is implemented on various parabolic PDEs from the literature including the forced heat equation and the advection-diffusion equation.
TL;DR: An error bound for the collocation method by spline in tension is developed for a nonlinear boundary value problem by using Sharp error bounds for the interpolating splines in tension in conjunction with recently obtained formulae for B-splines to develop an error bound depending on the tension parameters and net spacing.
Abstract: An error bound for the collocation method by spline in tension is developed for a nonlinear boundary value problemay″+by′+cy=f(·,y),y(0)=y0,y(1)=y1. Sharp error bounds for the interpolating splines in tension are used in conjunction with recently obtained formulae for B-splines, to develop an error bound depending on the tension parameters and net spacing. For singularly perturbed boundary value problems (|a|=e≪1), the representation of the error motivates a choice of tension parameters which makes the convergence of the collocation method problem at least linear. The B-representation of the spline in tension is also used in the actual computations. Some numerical experiments are given to illustrate the theory.
TL;DR: It is shown that the sequence of linear combinations Pn,n≥2l, is orthogonal with respect to a positive measuredσ if and only ifdσ is a Bernstein-Szegö measure anddμ is the product of a unique trigonometric polynomial and the Bernstein-Zegö measuredσ.
Abstract: Let \((P_n)\) and \(({\mathcal{P}}_n)\) be polynomials orthogonal on the unit circle with respect to the measures dσ and dµ, respectively. In this paper we consider the question how the orthogonality measures dσ and dµ are related to each other if the orthogonal polynomials are connected by a relation of the form \(\sum
olimits_{j = 0}^{k(n)} {\gamma _{j,n} {\mathcal{P}}_{n - j} (z)} = \sum
olimits_{j = 0}^{l(n)} {\lambda _{j,n} P_{n - j} (z)}\), for \(n \in {\mathbb{N}}\), where \(\gamma _{j,n} ,\lambda _{j,n} \in {\mathbb{C}}\). It turns out that the two measures are related by \(d\sigma \left( \phi \right) = {\mathcal{A}}\left( \phi \right)/{\mathcal{E}}\left( \phi \right)d\mu \left( \phi \right) + \sum M _j \delta \left( {e^{i\phi } - e^{i\kappa j} } \right)\) if \(l\left( n \right) + k\left( n \right) \leqslant n/3\), where \({\mathcal{A}}\) and \({\mathcal{E}}\) are known trigonometric polynomials of fixed degree and where the \(\kappa _j\)'s are the zeros of \({\mathcal{E}}\) on \(\left[ {0,\left. {2\pi } \right)} \right.\). If the \(l\left( n \right)\)'s and \(k\left( n \right)\)'s are uniformly bounded then (under some additional conditions) much more can be said. Indeed, in this case the measures dσ and dµ have to be of the form \({\mathcal{A}}\left( \phi \right)/{\mathcal{S}}\left( \phi \right)d\phi\) and \({\mathcal{E}}\left( \phi \right)/{\mathcal{S}}\left( \phi \right)d\phi\), respectively, where \({\mathcal{A}},{\mathcal{E}},{\mathcal{S}}\) are nonnegative trigonometric polynomials. Finally, the question is considered to which weight functions polynomials of the form \(\Phi _n : = \sum
olimits_{j = 0}^{l\left( n \right)} {\lambda _{j,n} P_{n - j} } + \sum
olimits_{j = 0}^{l\left( n \right)} {\gamma _{j,n} } P_{_{n - j} }^* ,\) where \(P_{_{n - j} }^* \left( z \right) = z^{n - j} \overline P _n \left( {1/z} \right)\) denotes the reciprocal polynomial of \(P_{n - j}\), can be orthogonal.
TL;DR: A very brief review is given of the striking way in which the Crank-Gupta model has enhanced the understanding of the well-posedness of free and moving boundary problems.
Abstract: A very brief review is given of the striking way in which the Crank-Gupta model has enhanced our understanding of the well-posedness of free and moving boundary problems.
TL;DR: The main result of this paper shows that this condition characterizes all configurations of Schoenberg-Whitney type such that in every neighborhood of T inK there must exist a configuration of T which admits unique Lagrange interpolation.
Abstract: Lagrange interpolation by finite-dimensional spaces of multivariate spline functions defined on a polyhedral regionK in ℝ
k
is studied. A condition of Schoenberg-Whitney type is introduced. The main result of this paper shows that this condition characterizes all configurationsT inK such that in every neighborhood ofT inK there must exist a configuration
$$\tilde T$$
which admits unique Lagrange interpolation.
TL;DR: A mathematical model to analyse some key aspects of the metal cast process is described involving the filling of the mould by liquid metal and simultaneously, undergoing both cooling and solidification (re-melting) phase change.
Abstract: A mathematical model to analyse some key aspects of the metal cast process is described involving the filling of the mould by liquid metal and simultaneously, undergoing both cooling and solidification (re-melting) phase change. A computational solution procedure based upon a finite volume discretisation approach, on both structured and unstructured meshes, is described. The overall flow solution procedure is based on the pressure correction algorithm SIMPLE suitably adapted to: (a) solve for the free surface with minimal smearing by the SEA algorithm, and (b) solve for the solidification/melting phase change using an enthalpy conservation algorithm developed by Voller, but with its root in the work of Crank many years ago.
TL;DR: Two algorithms for Chebyshev approximation of continuous functions on [0, 1]n using the modulus of continuity and the maximum norm estimated by a given finite data system are developed.
Abstract: In this paper, we develop two algorithms for Chebyshev approximation of continuous functions on [0, 1]n using the modulus of continuity and the maximum norm estimated by a given finite data system. The algorithms are based on constructive versions of Kolmogorov's superposition theorem. One of the algorithms we apply to neural networks.
TL;DR: In this article, the convergence of the GMRES/FOM and QMR/BiCG methods for solving nonsymmetric systems of equations was studied and it was shown that any type of residual norm convergence obtained using BiCG can also be obtained using FOM but on a different system of equations.
Abstract: We study the convergence of the GMRES/FOM and QMR/BiCG methods for solving nonsymmetric systems of equationsAx=b. We prove, in exact arithmetic, that any type of residual norm convergence obtained using BiCG can also be obtained using FOM but on a different system of equations. We consider practical comparisons of these procedures when they are applied to the same matrices. We use a unitary invariance shared by both methods, to construct test matrices where we can vary the nonnormality of the test matrix by variations in simplified eigenvector matrices. We used these test problems in two sets of numerical experiments. The first set of experiments was designed to study effects of increasing nonnormality on the convergence of GMRES and QMR. The second set of experiments was designed to track effects of the eigenvalue distribution on the convergence of QMR. In these tests the GMRES residual norms decreased significantly more rapidly than the QMR residual norms but without corresponding decreases in the error norms. Furthermore, as the nonnormality ofA was increased, the GMRES residual norms decreased more rapidly. This led to premature termination of the GMRES procedure on highly nonnormal problems. On the nonnormal test problems the QMR residual norms exhibited less sensitivity to changes in the nonnormality. The convergence of either type of procedure, as measured by the error norms, was delayed by the presence of large or small outliers and affected by the type of eigenvalues, real or complex, in the eigenvalue distribution ofA. For GMRES this effect can be seen only in the error norm plots.
TL;DR: The urgent need for adaptive solvers for these problems is pointed out and an ana posteriori error estimate for a model two-point hereditary boundary value problem is given.
Abstract: We give a brief indication of how elliptic, parabolic and hyperbolic partial differential equations with memory arise when modelling viscoelastic materials. We then point out the urgent need for adaptive solvers for these problems and, employing the methodology of Eriksson, Johnson et al. (e.g., SIAM J. Numer. Anal. 28 (1991)), we given ana posteriori error estimate for a model two-point hereditary boundary value problem. The strengths and weaknesses of the analysis and estimate are discussed.
TL;DR: A parallel algorithm for calculating theQR factorization of a banded system of linear equations using a systolic array processor is presented and an application to spline fitting is given.
Abstract: A parallel algorithm for calculating theQR factorization of a banded system of linear equations using a systolic array processor is presented and an application to spline fitting is given. The major advantage of the method is that the size of the processor array is fixed by the size of the bandwidth of the system to be solved. This allows the factorization of large systems using small systolic arrays. The cost of the method, in terms of storage and time, is optimal.
TL;DR: Two conditions on the signs of the coefficients of a bivariate polynomial which ensure that the zero set is a single curve are derived and will allow greater freedom when using algebraic curves for geometric modelling.
Abstract: Two conditions on the signs of the coefficients of a bivariate polynomial which ensure that the zero set is a single curve are derived. The first condition demands that all but one of the coefficients has the same sign. The second requires that the signs are ‘split’ by any straight line. Both properties are demonstrated by generalizing the set of isoparametric lines to a certain two-parameter family of curves. Equivalent curves are found for power, tensor-product Bernstein, exponential and triangular Bernstein polynomials. The second property will allow greater freedom when using algebraic curves for geometric modelling.
TL;DR: The paper reviews some recent theoretical work on the statistical properties of the distribution of concentration of an atmospheric pollutant, and focuses on the behaviour of the moments, particularly on the collapse of the skewness-kurtosis plot onto a parabola.
Abstract: The paper reviews some recent theoretical work on the statistical properties of the distribution of concentration of an atmospheric pollutant. This work is based on elementary physics. There is emphasis on the behaviour of the moments, particularly on the collapse of the skewness-kurtosis plot onto a parabola. Some of the consequences for modelling the probability density function of concentration are discussed, and examples of models are given.